202 
On Theories of Association 
(A) Barometric Data. 
The following table gives the value of tetrachoiic r t and of the coefficient of 
association Q*. 
Southampton and Laudale. 
Divisions 
Tetrachoric r t 
Yule's 
Association 
Divisions 
Tetrachoric r t 
Yule's 
Association 
29-15 
■6969 + -0378 
•9371 + -0144 
29-95 
•7991 + -0096 
•8868+ -0075 
29-25 
•7254 ±-0291 
•9335+ -0124 
30-05 
•7954+ -0100 
•8831 + -0075 
29S5 
•7476+ -0220 
■9244+ -0108 
30-15 
■7971+ -0110 
•8867 + -0075 
29-45 
•7410+ -0215 
•9067 + -0109 
30-25 
•8112+ -0127 
•9109+ -0070 
29-55 
•7303 ± -0177 
•8847 + -0111 
30-35 
•7983+ -0149 
•9229 + -0076 
29-65 
•7559+ -0140 
■8866 + -0097 
30-45 
•8300+ -0176 
•9579+ -0061 
29-75 
•7848+ -0114 
■8951 + -0083 
30-55 
■8452 + -0231 
•9761 + "0050 
29-85 
•7917+ -0102 
•8874 ±-0079 
* We have not thought it needful to deal in every case with both Mr Yule's coefficients. Since w is 
a function of Q, and its probable error is a function of the probable error of Q, there is no occasion to 
do so, and it opens a whole field for the display of statistical fallacy. Consider two quantities ?• and p 
defined in their mutual relations by r=(l- e)/(l + e) and p = (1 - e'/")/(l + e i /") ; then if we arrange a 
fourfold table so that r is positive, e will lie between 0 and 1 and p between 0 and 1. Clearly 
e = (l-r)/(l+r), e V»={l-p)l(l+p), 
or taking logarithmic differentials we find 
a /e = 2o-,./(l - f 1 ) and 
-^=2^/(1-^). 
Thus it follows that 
_p _ 
Now if 11 be positive r is always greater than p ; it follows accordingly that 1/p-p is greater than 
;ries 
1/c-r, but if we have any given series we can choose the value of n so great that <j ? \p is less than 
<r,./r. For if u 
0p//> 
then v -. 
" 1 
3/n 
As n becomes large, then the limit of 
eV* _ 1 
2« ' 
2/(6 
n 1 - e 2 '' 1 
and therefore Limit of » = 
which can be made as small as we please. 
Thus if we have any system of tetrachoric r t 's, it is always possible to devise a new system of p's, 
obtainable by the relations 
e=(l-r)/(l+r) and p= (1 - e 1 /")/(l +« V ")> 
which have a jp less than cr ,.//•. Thus, if we generalise Mr Yule, 0 = (1 - K 1 ' m )/(1 + K 1 ' ,t ) satisfies all his 
conditions of a coefficient of association, but we can always select n so that ff^/fi shall be less than 
o-qIQ or where Q and u> are Mr Yule's coefficients. Similarly we can deduce from the tetra- 
choric r t a definite function of it p, which has a ratio o^/p less than trjr. and yet satisfies all the 
so-called conditions of association. But for Mr Yule, and many others, the ratio of a quantity to 
its probable error is a measure of its significance. Hence by a proper choice of n we can always 
« 
